To analyze the transformation mapping of a 90-degree clockwise rotation about the origin, we can express this mathematically. A point \((x, y)\) after a 90-degree clockwise rotation transforms as follows:
\[ (x, y) \rightarrow (y, -x) \]
Now, let's consider the options provided:
a. Reflection over the line \( y = x \) transforms points as follows:
\[ (x, y) \rightarrow (y, x) \]
This is not equivalent to a 90-degree clockwise rotation.
b. Dilation with a scale factor of 1/4 transforms points as follows:
\[ (x, y) \rightarrow \left(\frac{x}{4}, \frac{y}{4}\right) \]
This does not change the angle of the points and is also not equivalent to a 90-degree clockwise rotation.
c. Translation shifts all points by a certain amount, transforming points as follows:
\[ (x, y) \rightarrow (x + a, y + b) \]
This also does not reorient points in a way that resembles rotation.
d. 270 degrees counterclockwise rotation transforms points as follows:
\[ (x, y) \rightarrow (-y, x) \]
This is equivalent to a 90-degree clockwise rotation due to the following reasoning: if you perform a 270-degree counterclockwise rotation, it is the same as a 90-degree clockwise rotation.
Thus, the equivalent transformation mapping for a 90-degree clockwise rotation about the origin is:
d. a 270 degrees counterclockwise rotation.